Bridges, channels and Arnold's invariants for generic plane curves (Q1862106)

Certain operations on self-transverse circle immersions to the plane which generalize the connected sum operation are defined. Also, the corresponding inverse process is studied and it is proved that any self-transverse circle immersion with \(n\) double points can be (non-uniquely) decomposed into at most \(n\) curves, each regularly homotopic, through self-transverse immersions, to one of three standard curves. (The three standard curves are the figure eight curve of tangential degree \(0\), the round circle of tangential degree 1 and the curve with one double point and tangential degree \(2\).) In [Adv. Sov. Math. 21, 33-91 (1994; Zbl 0864.57027)], \textit{V. I. Arnold} introduced three invariants \(J^+\), \(J^-\), and St of self-transverse circle immersions. The authors show how these invariants behave with respect to the operations mentioned above. This together with the decomposition result gives a new method for calculating Arnold's invariants.